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A sufficient condition for all short cycles. (English) Zbl 0882.05081
Summary: Generalizing a result of {\it R. Häggkvist}, {\it R. J. Faudree} and {\it R. H. Schelp} [Ars Comb. 11, 37-49 (1981; Zbl 0485.05038)], we prove that every non-bipartite graph of order $n$ with more than $(n-1)^2/4+ 1$ edges contains cycles of every length between 3 and the length of a longest cycle.

MSC:
 05C38 Paths; cycles
Full Text:
References:
 [1] Bondy, J. A.: Pancyclic graphs. J. combin. Theory ser. B 11, 80-84 (1971) · Zbl 0183.52301 [2] Bondy, J. A.: Large cycles in graphs. Discrete math. 1, 121-132 (1971) · Zbl 0224.05120 [3] S. Brandt, R.J. Faudree, W. Goddard, Weakly pancyclic graphs, submitted. [4] Dirac, G. A.: Some theorems on abstract graphs. Proc. London math. Soc. 3, 69-81 (1952) · Zbl 0047.17001 [5] Häggkvist, R.; Faudree, R. J.; Schelp, R. H.: Pancyclic graphs--connected Ramsey number. Ars combin. 11, 37-49 (1981) · Zbl 0485.05038 [6] Moon, J. W.; Moser, L.: On Hamiltonian bipartite graphs. Isr. J. Math. 1, 163-165 (1963) · Zbl 0119.38806